# Questions tagged [circulant-matrices]

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### Is the random point $(C,S)$ the same as $(1,0)+(\cos U,\sin U)=(1 + \cos U,\sin U)$, with $U$ a uniform r.v.?

Suppose that $\theta_1$ and $\theta_2$ are independent and identically distributed (i.i.d.) random variables and that $\theta_j$ has probability density function (PDF) $f_j = \frac{1}{2\pi}$ ($i.e.$, ...
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### Chromatic Polynomials of Circulant Graph With Two Parameters

I have been working with the chromatic polynomials of circulant graphs of prime order $p$ with two distinct parameters, i.e. $P_{p,i,j}(x):= P(C_{p}(i,j),x)$ with $1 \leq i \neq j \leq \ n/2.$ In ...
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Let $S$ be a finite subset of the complex unit circle and $1 \in S$. For each $n \in \mathbb N$, define $f_n\colon S^{n-1}\to\mathbb R$ by $$f_n(x) := \sum_{w^{n}=1}|x_1w+ x_2w^2\cdots+x_{n-1}w^{n-1}... 1answer 190 views ### Large submatrices of circulant matrices What properties of circulant matrices are inherited by their principal submatrices? To be more specific: Given a square (zero-one) matrix M of order n with all elements on its main diagonal ... 0answers 103 views ### Fullrankness of sum of time shifts I am working with finite Gabor frames and in this context a problem appeared which I am trying to solve for a couple of weeks now. Given a (p,k,1) cyclic difference set for \mathbb{Z}_p which is ... 0answers 141 views ### Lovasz theta and circulant graphs Let \theta(G) denote Shannon zero error capacity of graph G and \vartheta(G) be Lovasz upper bound for \theta(G). Let C_{2n+1} denote cycle graph with 2n+1 nodes. We know following two ... 1answer 273 views ### Bounds for maximum determinant of circulant matrices The Hadamard circulant conjecture states that there do not exist circulant Hadamard matrices with more than 4 columns. An n by n Hadamard matrix where the entries are chosen from \{-1,1\} ... 0answers 124 views ### About the properities of sum of powers of items in a polynomial Given a polynomial f_1=a_1x+a_2x^2+\cdots+a_{p-1}x^{p-1}\in\mathbb{Z}[x]/(x^{p}-1), with prime p, we may generate the other p-1 polynomials: \begin{eqnarray*} f_2&=&a_1^2x^2+\cdots+a_{p-... 0answers 63 views ### About the rank of a Pell equation-related matrix I have a question about the solution of Pell-equation over a prime field. I want prove that the matrix M is of rank \frac{p-1}{2}, with M=(m_{i,j})\in\left(\mathbb{Z}/(p^p-1)\mathbb{Z} \right)^{(... 1answer 364 views ### XOR circulant matrices? Take a function f: Z_N\rightarrow R. Construct an N \times N matrix where the (i,j)th element of the matrix is f(i-j), where i-j is interpreted mod Z_N. The resulting matrices are ... 1answer 368 views ### Partial Vandermonde circulant determinant expression Consider following partial Vandermonde type, circulant matrix \begin{bmatrix} x_1 & x_2 & 0 & \dots & 0 & x_n\\ x_1^2 & x_2^2 & x_3^2 & \dots & 0 & 0\\ \vdots ... 1answer 92 views ### Are certain normal matrices circulant? (Part 2) Let \mathcal{F} denote the family of real normal matrices A such that  A^TA=\begin{pmatrix} a & b \\ b & \ddots \end{pmatrix}, for b>0. As a user observed in the solution of Part 1 ... 2answers 368 views ### Is a normal matrix satisfying A^TA=… circulant? Let A=\{a_{ij}\} be a normal matrix such that a_{ij}\geq 0 with equality iff i=j. Suppose that$$ A^TA=\begin{pmatrix} a & b & \cdots & b\\ b & a & \ddots & \vdots\\ \...
If you are given a $0$-$1$ circulant matrix with $n$ rows and $n$ columns, is there an efficient way of determining if there exists a non-zero $\{-1,0,1\}$-vector in its kernel? Could this problem ...